a = {{2, 3}, {5, -1}, {12, 1.}}
b = {1, 7, -2}
{u, w, v} = SingularValueDecomposition[a]
u.w.Transpose[v]
Clear[x, y]; r = Thread[a.{x, y} == b]
f[x_] := If[x == 0, 0, 1/x]; SetAttributes[f, Listable];
w1 = f[Transpose[w]]
xx = v.w1.Transpose[u].b
d = 7; ContourPlot[Evaluate[r], {x, -d, d}, {y, -d, d}, 
 Epilog -> {Red, PointSize -> Large, Point[xx]}]